Abstract

In this paper we extend some of the computational results presented in [6] on finding an acyclic orientation of a graph which minimizes the maximum number of changes of orientations along the paths connecting a given subset of source-destination couples. The corresponding value is called rank of the set of paths. Besides its theoretical interest, the topic has also practical applications. In fact, the existence of a rank r acyclic orientation for a graph implies the existence of a deadlock-free routing strategy for the corresponding network which uses at most r buffers per vertex.

We first show that the problem of minimizing the rank is NP-hard if all shortest paths between the couples of vertices wishing to communicate have to be represented and even not approximable within an error in O(k1−ε) for any ε > 0, where k is the number of source-destination couples wishing to communicate, if only one shortest path between each couple has to be represented.

We then improve some of the known lower and upper bounds on the rank of all possible shortest paths between any couple of vertices for particular topologies, such as grids and hypercubes, and we find tight results for tori.

Keywords

Work supported by the EU TMR Research Training Grant N. ERBFMBICT960861, the EU ESPRIT Long Term Research Project ALCOM-IT under contract N. 20244, the French action RUMEUR of the GDR PRS and the Italian MURST 40% project “Algoritmi, Modelli di Calcolo e Strutture Informative”.